Horizontal and vertical integration in securities trading and settlement

Jens Tapking∗

and

Jing Yang∗∗

Working Paper no. 245

∗ European Central Bank.∗∗ Market Infrastructure Division, Bank of England.

The views expressed in this paper are the views of the authors and do not necessarily reflect thoseof the European Central Bank or the Bank of England. We thank Charles Goodhart, AndyHaldane, Harry Huizinga, John Jackson, Kevin James, Charles Kahn, Elias Kazarian, Will Kerry,Stephen Millard, Alistair Milne, Daniela Russo, Victoria Saporta, Hyun Song Shin, ElizabethWrigley, participants at the Bank of England’s Financial Stability Seminar, members of theSecurities Working Group of the Eurosystem and two anonymous referees for helpful commentsand discussions.

Copies of working papers may be obtained from Publications Group, Bank of England,Threadneedle Street, London, EC2R 8AH; telephone 020 7601 4030, fax 020 7601 3298,e-mail mapublications@bankofengland.co.uk

Working papers are also available at www.bankofengland.co.uk/wp/index.html

The Bank of England’s working paper series is externally refereed.

c Bank of England 2004ISSN 1368-5562

Contents

Abstract 5

Summary 7

1 Introduction 9

2 The model 15

3 Stock market equilibrium 17

4 Transaction price equilibrium 20

4.1 Pay-off functions 20

4.2 Complete separation 21

4.3 Vertical integration 23

4.4 Horizontal integration of CSDs 24

5 Welfare analysis 27

5.1 Net social benefit function and welfare maximum 27

5.2 Comparison of social benefits for different industry structures 28

6 Concluding remarks 29

Appendix A 31

Appendix B 37

References 49

3

Abstract

This paper addresses a very European issue, the consolidation of securities trading and settlement

infrastructures. In a two-country model, we analyse welfare implications of different types of

consolidation. We find that horizontal integration of settlement systems is better than vertical

integration of exchanges and settlement systems, but vertical integration is still better than no

consolidation. These findings have clear policy implications with regard to the highly fragmented

European securities infrastructure.

Key words: Securities trading and settlement, vertical and horizontal integration, substitutes and

complements.

JEL classification: G21, G15, L13.

5

Summary

Securities trading and settlement are essential parts of any securities transaction. Trading is the

process that results in an agreement between a seller and a buyer to exchange securities for funds.

Settlement refers to the actual transfer of securities from the seller to the buyer and the transfer of

the funds from the buyer to the seller. Trading is often carried out on securities exchanges,

whereas settlement of on-exchange trades takes place in entities called central securities

depositories (CSDs).

In the European Union, the securities trading and settlement infrastructure is highly fragmented.

There are over 20 national exchanges and about as many CSDs. Market participants, central banks

and regulators agree that consolidation is desirable. However, there is little agreement on what

kind of consolidation would be optimal. Some people prefer vertical integration, that is mergers of

exchanges with CSDs. Others favour horizontal integration of different exchanges or different

CSDs.

In this paper, we try to shed light on the pros and cons of the different types of consolidation in a

theoretical two-country model. There is an exchange and a CSD in each country. Investors can

buy and sell securities on both exchanges. All trades executed on a given exchange are settled in

the CSD of the same country. This reflects the current practice in all major markets. Hence, before

a security held in the CSD of country 1 can be offered on the exchange of country 2, it has to be

transferred to the CSD in country 2. This transfer is carried out through a so-called link, a

communication line between the two CSDs. A link transfer requires the services of both CSDs.

One CSD has to release the securities and the other CSD has to take them under custody.

We start from a general observation that has been well established in industrial economics. From

an economic welfare perspective, two goods that are substitutes should be supplied by different

decision-makers while two complements should be supplied by a single decision-maker. On the

basis of this observation, we argue as follows. The link services of the two CSDs are complements

since each securities transfer from one CSD to the other requires both services. Furthermore, the

link service of one CSD and the settlement service of the other CSD are complements since

transferring securities from one country to the other makes sense only if these securities are

afterwards traded and thus settled in the other country. The two CSDs should therefore be

7

operated by the same decision-maker (horizontal integration of CSDs).

However, this argumentation is valid only if the operating costs of the link are low enough to allow

the cross-border transfer of securities at reasonable costs. If the link is too expensive, a horizontal

integration of the CSDs is desirable only if it reduces these link-operating costs significantly. If it

does not reduce these costs, a vertical integration of the exchange and the CSD in each country is

preferable. This is because trading and settlement in a given country are also complements.

Furthermore, if there is no demand for foreign securities, there is also no demand for link transfers

regardless of whether the link-operating costs are high or low. In this case, the link has no

significance and the above argument in favour of horizontal integration of the CSDs is again not

valid. Instead, we again find that a vertical integration of the exchange and the CSD in each

country is preferable as trading and settlement in a given country are complements.

8

1 Introduction

The European securities trading and settlement infrastructure is highly fragmented. There are over

20 national exchanges and about as many central securities depositories (CSDs) in the EU. Market

participants, central banks and regulators agree that consolidation is desirable. However, there is

little agreement on what kind of consolidation would be optimal. Some people prefer vertical

integration, that is mergers of exchanges with CSDs (and clearing houses). Others favour

horizontal integration of different exchanges or different CSDs. In this paper, we try to shed some

light on the pros and cons of the different types of consolidation in a theoretical model.

In practice, all kinds of integration have been taking place in recent years. In Italy and, in a similar

way, in Germany, the exchange and the CSD have been merged to form a so-called vertical silo.

The merger of the exchanges of France, the Netherlands, Belgium and Portugal have created

Euronext. The CSDs of France, the Netherlands, the United Kingdom and Belgium have been

merged into Euroclear Group.

Securities exchanges and CSDs play essential roles in all major securities markets. Exchanges

help to match buyers and sellers of securities. CSDs are central store houses for securities. In most

countries, there is only one CSD and almost all securities issued under the country’s legislation are

stored there for their entire life - as physical papers or increasingly often electronically.

Furthermore, CSDs maintain records establishing ownership of securities. Major financial

institutions have securities accounts with the CSD and the account balances indicate the securities

owned by the respective financial institution (or its direct or indirect clients). Finally, CSDs act as

major settlement service providers: they organise the transfer of securities from a seller to a buyer.

If one financial institution sells securities to another, the transaction is settled by book entries in

the book of the CSD: the seller’s securities account with the CSD is debited and the buyer’s

securities account is credited.

Exchanges and CSDs co-operate closely. Most exchanges use, for reasons of costs or for legal

reasons, only one CSD to settle all trades executed on the exchange. All members of the exchange

have to have (directly or via an intermediary) securities accounts with that CSD. Whenever two

exchange members - a seller and a buyer - are matched on the exchange, the CSD receives

automatically from the exchange the instructions to debit the seller’s and to credit the buyer’s

9

securities account. This process is called straight-through processing (STP).

Special problems arise in case of cross-listed securities if the two exchanges on which the

securities are listed use different CSDs for settlement. Assume that an exchange A uses only CSD

A and another exchange B uses only CSD B. An investor may wish to sell on exchange B

securities held on his account with CSD A. Before he can do that, the securities have to be

transferred from CSD A to CSD B. For this purpose, CSDs maintain so-called (direct or indirect)

links. Only after the securities have been credited to an account with CSD B, can they be sold on

exchange B.

In this paper, we analyse the interactions between exchanges and CSDs in a two-country model.

There is an exchange and a CSD in both countries. There are two types of securities, country A

securities and country B securities. There are two types of investors, country A and country B

investors. Initially, all country A securities are held by country A investors on accounts with CSD

A and all country B securities are held by country B investors with CSD B. Initially, all investors

are members of their home exchange and CSD, but not of the foreign exchange and CSD. All

securities are listed on both exchanges. All trades executed on exchange A must be settled in CSD

A and all trades executed on exchange B must be settled in CSD B. The two CSDs maintain a link

so that securities can be transferred from one CSD to the other. Country A investors want to buy B

securities and country B investors want to buy A securities, so due to investors’ preferences only

trades between investors from different countries are possible. (1)

There are two ways to initiate transactions, for example between a country A investor who wants

to sell security A and a country B investor who wants to buy security A. First, the A investor

offers the securities on exchange A and the B investor orders them on exchange A. Settlement

takes place in CSD A and the link is not used. This is relatively costly for the B investor who

needs to become a member of exchange A and CSD A (directly or indirectly through an

intermediary). Second, the A investor transfers the securities through the link from CSD A to CSD

B and then offers them on exchange B while the B investor orders them on exchange B. This is

costly for the A investor who needs to transfer his securities through the link and must become a

(direct or indirect) member of exchange B and CSD B.

(1) This is a strong assumption. However, in Section 6, we note the effects of including investors’ demand for homesecurities.

10

Link transfers must be carried out jointly by the two CSDs. A crucial exogenous parameter of the

model is the operating cost of the CSDs for providing the link service. Each CSD sets a price the

investor has to pay for this service. Furthermore, each exchange sets a price for the execution of

trades and each CSD sets a price for the settlement of on-exchange trades. All four service

providers are operated by profit maximising firms. (2)

We analyse four different industry structures. (1) Under complete separation (CS), all four service

providers are operated by different independent firms and set their prices independently. (2) Under

vertical integration (VI), the exchange and the CSD in both countries are operated by the same

firm and thus co-ordinate their price-setting. (3) Under horizontal integration of the CSDs, both

CSDs are operated by the same firm. The exchanges are operated independently. We distinguish

two stages of horizontal integration: (a) purely legal integrations (LHI): though the CSDs are

operated by the same firm, they are technologically still different systems. The transfer of

securities through the link entails the same operating costs for the CSDs as under CS and VI. But

the CSDs set their prices for the link transfer as well as for the settlement of on-exchange trades in

a co-ordinated way. (b) Technical integration (THI): both CSDs are technologically merged into

one system so that a transfer of securities from one CSD to another does not entail any operating

costs so that the operating costs of the link are zero.

Horizontal integration of CSDs may indeed always lead eventually to THI. However, analysing

LHI is still not redundant since it helps to distinguish two effects of the transition from CS to THI.

This is a pure competition effect illustrated by the transition from CS to LHI. And a cost-reduction

effect illustrated by the transition from LHI to THI. Any kind of merger may have these two

effects. This positive cost-reduction effect may, however, be outweighed by a negative

competition effect. Analysing LHI as an intermediate step in the transition from CS to THI helps

to distinguish these two effects of horizontal integration of CSDs.

A welfare comparison of the four industry structures is the centre of our attention. The results of

this comparison are strikingly simple: VI and LHI, entail a (weakly) higher welfare then CS. That

is, the competition effects of the transition from CS to LHI and from CS to VI are positive. If the

link-operating costs under CS, VI and LHI exceed a certain threshold, then VI entails a higher

welfare than LHI, so that the competition effect is greater in the transition to VI than in the

(2) Some CSDs and exchanges are user-owned. In these cases, one may argue that the assumption of profitmaximisation is not realistic. However, many CSDs and exchanges are organised as profit maximising entities.

11

transition to LHI. If the operating costs of the link under CS, VI and LHI are smaller than this

threshold, then LHI entails a higher welfare then VI, ie the competition effect is greater in a

transition to LHI. However, THI always entails the highest economic welfare of all four structures.

In other words, even if the competition effect of a transition from CS to VI is greater than the

competition effect of a transition to THI, the overall welfare improvement is still greatest in case

of a transition to THI due to its cost-reduction effect.

Before we explain the economic reasons for these results, it is helpful to recall a finding from

basic industrial economics. (3) Consider a standard Bertrand duopoly. In this setting, a merger of

the two firms would decrease the economic welfare if the outputs of the firms are substitutes

(provided that the merger does not reduce production costs). However, if the outputs are

complements, then the merger would increase the welfare. The reason is the following: if two

firms produce substitutes and the price of both firms is relatively high, then one firm can easily

attract more demand by reducing its price a bit and boost up its profit. In equilibrium, both firms

therefore set relatively low prices. If the firms instead produce (perfect) complements, then the

demand at both firms depends on the sum of the prices of the two goods. Tourists for example

consider the sum of the prices for the flight to a holiday destination and for the accommodation

there. If the flight is cheap, they have high demand for hotel rooms even if these are relatively

expensive. If now both firms set a relatively low price, one firm would not lose too much demand

even if it increases its price significantly so that a higher price would result in a higher profit. In

equilibrium, both firms therefore set relatively high prices. However, if now both firms merge and

the new entity reduces the prices of the two complements, its profit would increase. Thus, the sum

of the prices of the two complements would be lower if the two firms are vertically integrated.

Looking again on our model, we find that the exchange and the CSD of the same country offer

perfect complements since trading on the exchange requires settlement in the CSD. This is why VI

entails a higher welfare then CS. Now compare LHI and VI. First, note that (trading and)

settlement in country A and (trading and) settlement in country B are substitutes. However, the

link service provided by CSD A and the link service provided by CSD B are perfect complements.

From that perspective, it is not immediately clear whether LHI or VI leads to a higher welfare.

The reason why VI leads to a higher welfare than LHI if the link-operating costs are high is

simple. In this case, transferring securities through the link is too costly, ie the link is not used and

(3) See for example Tirole (1988) and Shy (1996).

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securities are always traded where they are initially held. Thus, the CSDs in fact do not compete at

all, neither in substitutes nor in complements. CS and LHI now lead exactly to the same

equilibrium. However, VI leads to lower prices than CS and LHI and thus to a higher welfare

because the exchange and the CSD of the same country offer perfect complements. The reason

why LHI leads to a higher welfare than VI if the link-operating costs are low is a bit more

complex and will be discussed in detail later. (4)

Finally, it is clear that THI leads to a higher welfare than LHI due to cost reductions. Thus, it is

clear that THI is the best if the link-operating costs are low since LHI is better than CS and VI in

this case. However, if the link-operating costs are high, then LHI is not better anymore than VI.

But now, the cost-reduction effect of THI is even more significant and THI is still better than all

other industry structures.

Our findings have obvious implications for the policy discussion on what kind of consolidation

may be most desirable for the European securities trading and settlement infrastructure. However,

it is important to draw attention to two important limitations of our model. First, we assume that

the exchanges cannot choose which CSD they use. Each exchange has to settle on the CSD

located in the exchange’s country. This assumption clearly reduces the potential competition

between the two CSDs significantly. If under CS, the CSDs were forced to compete with each

other for the exchanges, then any type of horizontal integration of the CSDs may result in a

negative competition effect. Currently, national exchanges are to a large extent bound to use

exclusively the respective national CSD so that our assumption seems to be realistic enough.

However, this may change at some point and it would be interesting to analyse the implications of

such a change. Second, we do not allow for OTC trading, ie assume that trading exclusively takes

place on an exchange. Though this may appear realistic for equities, it is less realistic for bonds.

There is a large body of literature on vertical integration. An overview is in Perry (1989). Most

applications of the theory of vertical integration are on network industries in which a monopolistic

upstream firm (supplier of a network) produces an essential input for several competing

downstream firms (users of the network). The main issue is the implications of a vertical merger

of the upstream firm with one of the downstream firms. A prominent example is Vickers (1995).

However, our study largely differs from this literature in the following aspects. There is no

(4) See the two paragraphs after proposition 7.

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analogy of downstream/upstream firms in a trading and settlement system. Exchange do not

receive services from and do not make payments to CSDs and the other way round. Furthermore,

there is only one upstream firm in network industries while we have two exchanges and two CSDs

in our model.

Another body of literature looks at competition among interconnected networks like two operators

of mobile phone networks. See for examples Laffont, Rey and Tirole (1998a and 1998b) and

Laffont and Tirole (1996). The situation analysed by this literature somehow resembles the

competition between two CSDs connected by a link. However, the literature on competition

among interconnected networks focuses mainly on access price regulation while vertical

integration is not an issue.

There is some literature on competition, co-operation and consolidation of securities exchanges.

Examples are Domowitz (1995) and Shy and Tarkka (2001). A general discussion of competition

among exchanges with a special view to Europe is Di Noia (1998). In this type of literature,

exchanges are often considered as networks and consolidation of different networks as a way to

pool liquidity in one place. (5) There is currently very little literature on competition and

consolidation of different settlement service providers. Examples are the empirical paper by

Schmiedel, Malkamaki and Tarkka (2002), the theoretical work of Holthausen and Tapking

(2003), Kauko (2003) and Koeppl and Monnet (2003). Kauko (2003) is related to our paper in that

it looks at links as devices that create competition between CSDs. Koeppl and Monnet (2003) is to

our knowledge the only other paper that looks at exchanges and CSDs in one model. They show in

a two-country model with an exchange and a CSD in each country that due to asymmetric

information, an efficient merger of the CSDs is difficult to achieve, if initially the exchange and

the CSD are integrated in both countries.

The paper is organised as follows: Section 1 describes the assumption of our model. Since our

model is a two-stage model, we analyse the two stages in turn in Sections 2 and 3. Section 2 looks

at the behavior of investors for given prices of the exchanges and CSDs. Section 3 analyses the

price-setting behavior of the exchanges and CSDs under the four different industry structures

described above. Finally, the welfare implications of the different industry structures are

determined in Section 4.

(5) On competition among networks, see Economides (1996, 2003), Economides and Schwartz (1995).

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2 The model

There are two countries A and B. There is a stock exchange in each country. Furthermore, there is

a CSD in both countries. All trades on the stock exchange of country A are settled in the CSD of

country A and all trades on the stock exchange of country B are settled in the CSD of country B.

For each trade executed on exchange A and then settled in CSD A, both the seller and the buyer

have to pay pTA to stock exchange A and pSA to CSD A. Similarly, the prices in country B are pTBand pSB . The exchanges’ marginal costs for executing a trade are cT and the CSDs’ marginal costs

for settling a trade executed on an exchange are cS.

The two CSDs maintain a bilateral (direct) link which can be used to transfer securities from one

CSD to the other. For each transfer, CSD A charges a price qA and CSD B charges qB . Each CSD

incurs marginal costs of cL for such transfers.

In each country, there is a set [0, 1] of investors. Each country A investor is a member of stock

exchange A and of CSD A (ie has a securities account in CSD A). Similarly, each country B

investor is a member of stock exchange B and of CSD B (ie has a securities account in CSD B).

Initially, no country A investor is a member of exchange B or CSD B and no country B investor is

a member of exchange A or CSD A. However, any investor can decide to become a (direct or

indirect) member of the foreign stock exchange and CSD to be able to trade there. For simplicity,

we assume that the exchanges and CSDs do not ask for a fee for this remote access. However, the

(exogenous) costs for establishing remote access - tT for access to the respective exchange and tSfor access to the respective CSD - are borne by the investor. These costs refer to, for example, IT

facilities an investor needs to set up and maintain to have remote access. We define t = tT + tS asthe overall costs for remote access. (6)

There are two stocks. Stock A has been issued into CSD A and stock B has been issued into CSD

B. Both stocks are listed (and quoted) on both exchanges and thus eligible for settlement in both

CSDs. Initially, each country A investor owns one share of stock A which is kept on his account

with CSD A. Each country B investor owns one share of stock B which is kept on his account

with CSD B. Country A investors can sell stock A and buy stock B and country B investors can

sell stock B and buy stock A. However, buying or selling abroad requires that the investor first

(6) If an investor is member of the foreign exchange and CSD indirectly through an intermediary, the investor wouldpay prices for trading and settlement abroad via its intermediary.

15

becomes a member of the foreign exchange and CSD. Moreover, selling abroad requires that the

investor first transfers his share to the foreign CSD.

An investor’s benefit of holding stocks depends on the location of the investor: For A investor

i ∈ [0, 1], the benefit of holding one share of stock A is i and the benefit of holding one share ofstock B is 1− i . For B investor i ∈ [0, 1], the benefit of holding on share of stock B is i and thebenefit of holding one share of stock A is 1− i . Each investor has nine alternatives given by thefollowing table:

Alternative 1 Do not trade

Alternative 2 Sell home stock at home, do not trade foreign stock

Alternative 3 Sell home stock abroad, do not trade foreign stock

Alternative 4 Do not trade home stock, buy foreign stock at home

Alternative 5 Do not trade home stock, buy foreign stock abroad

Alternative 6 Sell home stock at home, buy foreign stock at home

Alternative 7 Sell home stock at home, buy foreign stock abroad

Alternative 8 Sell home stock abroad, buy foreign stock at home

Alternative 9 Sell home stock abroad, buy foreign stock abroad

Let pA ≡ pTA + pSA, pB ≡ pTB + pSB and q ≡ qA + qB . For country A investor i ∈ [0, 1], thebenefits from each alternative are given by the following table:

1 u1i,A = i2 u2i,A = rAA − pA3 u3i,A = rAB − t − q − pB4 u4i,A = i + 1− i − rBA − pA5 u5i,A = i + 1− i − rBB − t − pB6 u6i,A = rAA + 1− i − rBA − 2pA7 u7i,A = rAA + 1− i − rBB − t − pA − pB8 u8i,A = rAB + 1− i − rBA − t − q − pA − pB9 u9i,A = rAB + 1− i − rBB − t − q − 2pB

Whenever we use in this paper symbols with subscripts consisting of two letters, each either A or

B (eg rAA, xBA, SBB), then the first letter refers to a stock and the second to a country. Here, rAA is

the price of stock A in country A, rBA is the price of stock B in country A and rBB and rAB denote

the respective prices in country B. If we replace in the table the index A by B and the index B by

16

A wherever A and B occur, then we get the benefits of country B investor i ∈ [0, 1] for each of hisnine alternatives. Note that there are economies of scope in international trading. As long as

pA = pB , rAA ≤ rAB and rBB ≤ rBA, an investor who has made the investments t + q necessary tosell abroad will always buy abroad as well (u9i,A > u8i,A).

Decisions are taken in two steps. First, the two exchanges and the two CSDs simultaneously set

the transaction prices pTA , pSA, pTB , pSB , qA and qB . Second, each investor selects one alternative out

of his nine alternatives to maximise his benefit given all six transaction prices and all four stock

prices. Simultaneously, the four stock prices take values such that all four stock markets are in

equilibrium.

Note that our model covers different arrangements for cross-border trading and settlement since

we allow for direct remote access to the foreign exchange and CSD as well as indirect remote

access through an intermediary. However, it is assumed that all trades are executed on an

exchange. In other words, we assume that there is no international custodian bank that could

internalise international buy and sell orders without routing them to an exchange.

3 Stock market equilibrium

In this section, we discuss the stock market equilibrium for given transaction prices. Determining

the stock market equilibrium is quite a cumbersome exercise. First, for given stock prices rAA,

rBA, rBB and rAB and given transaction prices pTA , pSA, pTB , pSB , qA and qB , the demand and supply

functions for stock A in country A (SAA and DAA), for stock A in country B (SAB and DAB), for

stock B in country A (SBA and DBA) and for stock B in country B (SBB and DBB) have to be

determined. Details can be found in Appendix A. Second, for given transaction prices pTA , pSA, pTB ,

pSB , qA and qB , the stock market equilibrium has to be found. A stock market equilibrium for given

transaction prices is defined as a constellation of stock prices rAA, rBA, rBB and rAB such that

SAA = DAA = xAA, SAB = DAB = xAB , SBA = DBA = xBA and SBB = DBB = xBB . Here, xAA,xAB , xBA and xBB are the equilibrium trading volumes in the four stock markets.

However, we show in Appendix B that with this definition, there are under some parameter

constellations multiple stock market equilibria with different trading volumes. For this reason, we

apply the following refinement: If pA = pB , there may be two equilibria - one characterised by

17

xBB = xAB = 0 and another by xAA = xBA = 0. In this case, we select the former if pA < pB andthe latter if pA > pB . If pA = pB , there may be infinitely many equilibria. In this case, we selectthe one characterised by xAA = xBB and xAB = xBA. This refinement ensures that we have toconsider exactly one equilibrium for most parameter constellations. The equilibrium is given by

the following two propositions.

Proposition 1 If t ≥ q, then the stock market equilibrium for given transaction prices ischaracterised by the following trading volumes:

I) If pA < pB , then

xAA = xBA = max{12 − 14(t + q)− pA, 0}, xAB = xBB = 0.

II) If pB < pA, then

xBB = xAB = max{12 − 14(t + q)− pB, 0}, xBA = xAA = 0.

III) If pA = pB , then

xAA = xBA = xAB = xBB = max{14 − 18(t + q)− 1

2 pA, 0}.

If t is relatively high and q relatively low, then the link is relatively cheap and becoming a member

of the exchange and the CSD of the foreign country is relatively expensive. In this case, we find

that the investors are very sensitive regarding the difference between pA and pB . As soon as these

prices are not equal, all stocks are transferred via the link from the country with the higher to the

country with the lower trading and settlement price and all trade takes place in the country with

the lower price. Ie trading and settlement in country A and trading and settlement in country B are

perfect substitutes.

Note that if pA = pB , all linear combinations of the trading volumes under I) and II) wouldcharacterise an equilibrium if we did not use the above refinement. Since we have

xAA = xBA = xAB = xBB for pA = pB , half of the trading country A investors sell in country Band half of the trading country B investors sell in country A. These are the investors who use the

18

link. Furthermore, it follows that these and only these investors buy abroad. Ie every trading

investor either uses the link by himself or trades with an investor who uses the link. And those

investors who do not use the link do not become members of the foreign exchange and the foreign

CSD. In other words, for given trading volumes, remote membership is created as little as possible

and the link is used as much as possible. In equilibrium, we find that rAB = rBA > rAA = rBB , iethe investors who use the link are compensated by favourable stock prices for the additional

transaction costs they bear. This also implies that the investors exploit the economies of scope

mentioned in the previous section: no investor will ever sell abroad and buy at home.

The situation looks very different in case that t ≤ q:

Proposition 2 If t ≤ q, then the stock market equilibrium for given transaction prices ischaracterised by the following trading volumes:

I) If |pA − pB| ≤ 12(q − t), then

xAA = max{1−t2 − pA, 0}, xBB = max{1−t2 − pB, 0}, xBA = xAB = 0.

II) If pB − pA ≥ 12(q − t), then

xAA = max{1−t2 − pA, 0}, xBA = max{1−q2 − pA, 0}, xAB = xBB = 0.

III) If pA − pB ≥ 12(q − t), then

xBB = max{1−t2 − pB, 0}, xAB = max{1−q2 − pB, 0}, xBA = xAA = 0.

If t is relatively low and q relatively high, the link is relatively expensive and becoming a member

of the foreign exchange and CSD is relatively cheap. In this case, we find that xAA and xBB are

positive and xAB = xBA = 0, if the difference between pA and pB is moderate. Ie investors fromboth countries become members of the respective foreign exchange and CSD and the link is not

used. Stock A is traded only in country A and stock B is traded only in country B. Only if the

difference between pA and pB is sufficiently high, all stocks are transferred via the link from the

country with the higher to the country with the lower trading and settlement price and all trade

19

takes place in the country with the lower price. Ie trading and settlement in country A and trading

and settlement in country B are now imperfect substitutes. (7)

4 Transaction price equilibrium

After determining the stock market trading volumes in equilibrium for given transaction prices, we

now look at the first stage of the model. Here, the transaction prices are set by the two exchanges

and the two CSDs.

4.1 Pay-off functions

To begin with, we define the pay-off functions of the players. The profit function of exchange A is

defined by

π TA = 2(xAA + xBA)(pTA −12cT )

and for exchange B, we get

π TB = 2(xBB + xAB)(pTB −12cT )

Note that an exchange receives the price pTA or pTB twice for each trade executed because both the

seller and the buyer have to pay the price. This is why the trading quantities are multiplied by 2.

However, cT is defined as the costs of the exchange for executing a trade.

For the CSDs, we get

π SA = 2(xAA + xBA)(pSA −12cS)+ (xAB + xBA)(qA − cL)

and

π SB = 2(xBB + xAB)(pSB −12cS)+ (xAB + xBA)(qB − cL)

The first term refers to the profits from settling trades on the respective exchange, while the second

term refers to profits from operating the link.

Since we are going to look at different industry structures, we define π A = π TA + π SA andπ B = πTB + π SB as the profit of a company operating both the exchange and the CSD in therespective country (vertical integration of exchange and CSD). The profit of a company operating

both CSDs would be π S = π SA + π SB (horizontal integration in settlement).(7) Note that 2 describes two equilibria for |pA − pB | = 1

2 (q − t). In fact, all linear combinations of these twoequilibria are also equilibria in this case. We have omitted a detailed description of this case, because it has no impacton the further analysis. Furthermore, note that for t = q, the two propositions describe two different equilibria.

20

In the following three sections, we analyse the equilibrium transaction prices and trading volumes

under different industry structures. However, before we enter into the analysis, a few things should

be noted.

First, for any given parameter constellation, there is a multiplicity of no-trade equilibria which we

will ignore. Assume that both exchanges set prices that are so high that the demand for trade

would be zero even if both settlement (and link) prices are zero. It is clear that any settlement

prices are best responses of the operators of the CSDs. Analogously, assume that both CSDs set

settlement prices that are so high that the demand for trade would be zero even if both trading

prices are zero. Now any trading prices are best responses of the exchanges. Thus, there are

always equilibria with no trade and prohibitively high prices for trading and settlement. Since

such equilibria describe extreme co-ordination failures and characterise hardly interesting trading

allocations, we ignore them.

Second, under certain circumstances there are equilibria in which the marginal costs for trading

and/or for the settlement of trades are higher than the respective prices (ie pSA <12cS and/or

pSB <12cS and p

TA <

12cT and/or p

TB <

12cT ).

(8) We ignore these equilibria, too.

Finally, to avoid corner solutions in equilibrium, we assume α ≥ 0 and β ≥ 0 with

α ≡ 1− t − cS − cTβ ≡ 1− 1

2t − cS − cT − cL

4.2 Complete separation

To begin with, we look at an industry in which all four service providers are operated

independently by different companies. We concentrate on symmetric equilibria only, ie equilibria

with pSA = pSB , pTA = pTB and qA = qB . The following proposition describes an equilibrium inwhich the link is used.

(8) This is possible because companies that offer more than one service, for example settlement and link services or -in case of vertical integration - trading and settlement services, can cross-subsidise the different services.

21

Proposition 3 If t − 2cL > 2α, then there is one and only one symmetric equilibrium in whichthe link is used.

(1) If t − 2cL > 4α, it is characterised by

pTA = pTB =12cT , pSA = pSB =

12cS, qA = qB = 23β + cL

xAA = xBA = xAB = xBB = 112β

(2) If 4α > t − 2cL > 2α, then it is characterised by

pTA = pTB =12cT , pSA = pSB =

12cS, qA = qB = 12 t

xAA = xBA = xAB = xBB = 14α

If t − 2cL < 2α, then there is no symmetric equilibrium in which the link is used.

First note that only if the operating costs cL of the link are sufficiently low, there is an equilibrium

in which the link is used. Note also that in this equilibria, both the trading and the settlement prices

are equal to marginal costs. The reason is that if cL is low, so then are the prices qA and qB for the

link and it is cheap to substitute trading and settlement in one country for trading and settlement in

the other country. The exchange and the CSD in country A enter into perfect price competition

with the exchange and CSD in country B. This leads to a situation in which the prices equal

marginal costs. Note that part (2) of the proposition describes corner cases with qA = qB = 12 t .

According to proposition 3, an equilibrium in which the link is used exists only if t − 2cL > 2α.However, there is always an equilibrium in which the link is not used:

Proposition 4 There are always symmetric equilibria in which the link is not used. The set of all

22

such equilibria is characterised by

pTA = pTB =16(1− t − cS + 2cT ), pSA = pSB =

16(1− t − cT + 2cS),

qA = qB ≥ 12 t

xAA = xBB = 16α, xBA = xAB = 0

One might not be surprised to find an equilibrium in which the link is not used, if cL is high. It is

also not a surprise that in such an equilibrium, trading and settlement prices are higher than

marginal costs since there is no (direct) competition between the two countries if cL and thus the

link prices are high. However, why are there equilibria in which the link is not used, if cL is low?

The reason is a simple co-ordination problem: if for example qA is, say, higher than t , then the link

will not be used no matter how low qB is. CSD B has therefore no reason not to set qB > t . For

the same reasons, CSD A has now no reason not to set qA > t . Thus, qA = qB ≥ t alwaysconstitutes an equilibrium.

Thus, we do not have a unique equilibrium, if t − 2cL > 2α. However, there are good reasons toselect in this case the equilibrium described in proposition 3. First, it is easy to show that the

CSDs reach a higher profit in the equilibrium described in proposition 3 than in the one of

proposition 4. Second, we have argued in the previous subsection that we do not consider

equilibria in which there is no trade at all due to co-ordination failures. For the same reasons, we

can ignore the equilibria of proposition 4 in case that t − 2cL > 2α. Ie from now on, we assumethat the CSDs co-ordinate on the equilibrium described in proposition 3 whenever t − 2cL > 2α.

4.3 Vertical integration

We now assume that in both countries the CSD and the exchange are operated by the same firm.

The operator of the silo in country A sets pSA, pTA and qA, the operator of the other silo

simultaneously sets pSB , pTB and qB . Again, we concentrate on symmetric equilibria only, ie

equilibria with pSA = pSB , pTA = pTB and qA = qB and find:

Proposition 5 Proposition 3 holds also under vertical integration.

23

The economic intuition for this result is of course the same as for proposition 3. If cL is low, it is

cheap to transfer securities from one country to the other. The operator of the exchange and the

CSD in country A and the operator of those in country B enter into perfect competition so that

trading and settlement fees go down to marginal costs.

As under complete separation, there is always an equilibrium in which the link is not used:

Proposition 6 There are always symmetric equilibria in which the link is not used. The set of all

such equilibria is characterised by

pTA + pSA = pTB + pSB =14(1− t + cS + cT ),

qA = qB ≥ 12 t

xAA = xBB = 14α, xBA = xAB = 0

Note that the trading and settlement prices are lower in the equilibrium of proposition 6 than in the

one of proposition 4. The reason is that the CSD and the exchange of a given country offer

complements. As explained in the introduction, mergers of firms that offer complements reduce

prices.

Again, we do not have a unique equilibrium if t − 2cL > 2α. But it is again easy to see that theCSD’s profits are higher in the equilibrium described in proposition 5 than in the one of

proposition 6. For that reason and because proposition 6 describes equilibria that are due to

co-ordination failures if cL is low, we assume again that the CSDs co-ordinate on the equilibrium

described in proposition 5 whenever t − 2cL > 2α.

4.4 Horizontal integration of CSDs

We now look at a horizontally integrated structure, ie the two CSDs are operated by one company.

In this case, we have three players. The operator of exchange A sets pTA and has pay-off function

π TA. The operator of exchange B sets pTB and has pay-off function πTB . Finally, the operator of the

two CSDs sets pSA, pSB , qA and qB; his pay-off function is π S. We assume that the operator of the

CSDs cannot price-discriminate between the two countries, ie he has to set pSA = pSB . This

24

assumption can be justified by the existence of competition authorities like the European

Commission in the EU that would not allow price discrimination. We first consider a purely legal

integration of the CSDs. Concentrating only on symmetric equilibria (pTA = pTB), we get

Proposition 7 (1) For t − 2cL > 0, the set of all symmetric equilibria is characterised by

pTA = pTB =12cT , q + 4pSA = 1−

12t + cS + cL − cT , pSA = pSB , q ≤ t

xAA = xBA = xAB = xBB = 18β

(2) For t − 2cL < 0, the set of all symmetric equilibria is characterised by

pTA = pTB =16[1− t + 2cT − cS], pSA = pSB =

16[1− t + 2cS − cT ],

q ≥ t

xAA = xBB = 16α, xBA = xAB = 0

If cL is low, ie t − 2cL > 0, then q is low and the two exchanges enter into perfect pricecompetition that leads to a situation in which the trading prices are equal to marginal costs.

However, the prices for settlement and for the link are not unique, but only the weighted sum

q + 4pSA of both. The reason is subtle, but important to note. Remember that there are economiesof scope in our model: if an investor decides to sell abroad, he will buy abroad as well. If he is for

example a country B investor, then the transaction price he has to pay to the integrated CSDs is

q + 2pSA, the country A investor he sells to pays pSA and the country A investor he buys from alsopays pSA. The total transaction price paid by investors to the CSDs is thus q + 4pSA. If now qincreases and 4pSA decreases by the same amount so that q + 4pSA remains unchanged, then the Binvestor pays more and the two investors from country A pay less. However, it can be shown that

now, rAB increases and rBB decreases until the B investor is compensated for that. Thus, if q

increases and q + 4pSA remains unchanged, the investors’ equilibrium behaviour does not change.

We still look at t − 2cL > 0 and compare the prices for settlement and for the link given by thepropositions 7, 3 and 5. The settlement services of the two CSDs are substitutes, while their link

services are complements. For that reason, one would expect that pSA and pSB are higher under

25

horizontal integration, while q is higher under complete separation and under vertical integration.

However, since the prices for settlement and for the link are not unique under horizontal

integration, we can only compare the weighted sum q + 4pSA. It is easy to show that q + 4pSA ishigher under complete separation and under vertical integration than it is under horizontal

integration. The reason is closely related to what was explained in the previous paragraph.

Investors’ choice only depends on the weighted sum q + 4pSA. That reveals a relation between thetwo settlement services on the one hand and the two link services on the other: they are perfect

complements. Using the link is beneficial for investors only if they afterwards use the settlement

service of one CSD. And the settlement service of a CSD is used only if the link services are used

before. This is due to the economies of scope described above. As the settlement services and the

link services are complements, prices are lower if all these services are provided by the same

player, ie under horizontal integration.

Finally, if t − 2cL < 0, then the link is too expensive to be used and purely legal horizontalintegration and complete separation lead to the same results.

Now consider a complete technical integration of the CSDs. Ie the two CSDs are operated on the

same system and the operating costs of the link are thus cL = 0, while tS = 0 as well. Theequilibrium for this case now follows directly from part (1) of proposition 7:

Proposition 8 For cL = tS = 0 (THI), the set of all symmetric equilibria is characterised by

pTA = pTB =12cT , q + 4pSA = 1−

12tT + cS − cT , pSA = pSB

xAA = xBA = xAB = xBB = 18[1−12tT − cS − cT ]

It should be mentioned here that a THI may entail substantial costs that are to be borne by the

operator of the integrated CSDs. Since we ignore in our model these costs, proposition 8 describes

a longer-term equilibrium while in the short run, the operator of the CSD may add a surcharge to

its prices to cover the cost of THI.

26

5 Welfare analysis

We now compare the welfare characteristics of the different industry structures discussed above.

We first determine the general net social benefit function and describe the general welfare

maximum. We than calculate the net social benefits for the different industry structures.

5.1 Net social benefit function and welfare maximum

We start with the net benefits from trade of the country A investors and of the country B investors.

Country A investors sell a volume of xAA + xAB of stock A to country B investors and buy avolume of xBB + xBA of stock B from country B investors. Those country A investors who do notsell have a benefit of holding stock A of

1

xAA+xAB

i di

Those countryA investors who sell stock A have a benefit of selling it of

xAA(rAA − pA)+ xAB(rAB − pB − q)Those country A investors who buy stock B have a benefit of doing this of

xBB+xBA

0

(1− i) di − xBB(rBB + pB)− xBA(rBA + pA)

On top of that, country A investors who trade in country B must get connected to country B. Note

the following: each investor who wants to trade in country B needs to get connected to country B

only once, even if he trades both stocks in country B. Furthermore, a situation in which some

country A investors trade in country B only stock A and some other country A investors trade in

country B only stock B is not possible. Ie the overall number of country A investors who get

connected to country B is max{xBB, xAB} and the costs for country A investors from gettingconnected to country B are thus given by t max{xBB, xAB}. The net benefit of country A investorsis given by the sum of these three first components minus t max{xBB, xAB}, ie by

BA = 12− 12(xAA + xAB)2 − 12(xBB + xBA)

2 + (xBB + xBA)−t max{xBB, xAB} + xAA(rAA − pA)+ xAB(rAB − pB − q)−xBA(rBA + pA)− xBB(rBB + pB)

27

We get the net benefits BA of country B investors in a similar way. The net benefits of the

economy as a whole is given by

B = BA + BB + π TA + π TB + π SA + π SB= (1− cS − cT )[xAA + xAB + xBB + xBA]− (xAA + xAB)2 − (xBB + xBA)2

−2cL(xAB + xBA)− t max{xBB, xAB} − t max{xAA, xBA} + 1

It is now easy to determine the welfare maximum, ie the maximiser of the function B. It is given

by

Proposition 9 If 2cL < t , then the welfare maximum is obtained with

xAA = xBB = xAB = xBA = 14β. If 2cL > t , then it is obtained with xAA = xBB = 1

2α,

xAB = xBA = 0.

Note that it is optimal to have trading of both securities in both countries and thus to use the link if

the operating costs of the link are relatively low (2cL < t). Given the relations described in the

propositions 1 and 2, these welfare maximising trading volumes can be implemented with the

prices pA = pB = 12cT + 1

2cS and q = 2cL (prices equal to marginal costs).

5.2 Comparison of social benefits for different industry structures

Taking now the results from the previous section, it is easy to calculate the net social benefits for

the different industry structures. For CS, it is given by

BCS(cL) =

1+ 5

18β2, if t − 2cL > 4α

1+ αβ − 12α2, if 4α > t − 2cL > 2α

1+ 518α

2, if 2α > t − 2cLIn a similar way, we find for VI

BV I (cL) =

1+ 5

18β2, if t − 2cL > 4α

1+ αβ − 12α2, if 4α > t − 2cL > 2α

1+ 38α2, if 2α > t − 2cL

28

For LHI, we get

BLH I (cL, t) = 1+ 3

8β2, if t − 2cL > 0

1+ 518α

2, if 0 > t − 2cLFinally, we have

BT H I = BLH I (0, t = tT ) = 1+ 38[1−12tT − cS − cT ]2

The comparison of the net social benefits is straightforward and given by

Theorem 10 If t > 2cL , then BT H I > BLH I > BV I = BCS. If 2cL > t , thenBT H I > BV I > BLH I = BCS.

To understand this result, we only have to compare equilibrium prices since it is clear that the

lower the equilibrium prices the higher the social benefits. The intuition for the differences in

equilibrium prices for the different industry structures has been discussed already in Section 4. If

t > 2cL , then there is a strong complementary relation between the services provided by the two

CSDs. The two link services are complements. And the link service of one and the settlement

service of the other CSD are complements. The prices are therefore lower if all these services are

offered by the same player and BLH I > BV I = BCS. If t < 2cL , then the link is too expensive tobe used. What matters are the settlement services of the CSDs and these are neither substitutes nor

complements if t < 2cL . Since trading and settlement within a country are complements, the

prices for these services are lower if they are offered by the same player so that we get

BV I > BLH I = BCS. Finally, if 2cL > t , then the cost-saving effect of THI is so high thatBT H I > BV I . (9)

6 Concluding remarks

In this paper, we have analysed the welfare implications of different structures of the securities

trading and settlement industry in a two-country model. The result of our analysis is remarkably

simple: complete horizontal integration of CSDs leads to higher welfare than vertical integration

of exchanges and CSDs, and vertical integration leads to higher welfare than complete separation.

However, it should be emphasised that this paper is only a first step to analyse a complex question.

More research is needed to get the complete picture and final policy conclusions should not yet be(9) Assume that LHI already leads to a cost-saving effect so that it reduces the link-operating costs to cL < cL . If2cL ≤ t , then LHI would now outperform VI even if t < 2cL . Thus, horizontal integration is already the best type ofconsolidation if it makes the economic cost of the link smaller than the cost of remote access.

29

drawn. In this context, two limitations of the model mentioned already in the introduction need to

be emphasised again. First, we assume that the CSDs cannot compete for the exchanges, that is

each exchange is forced to settle all trades in the domestic CSD. And second, we do not allow for

OTC trading (including internalisation of trades by an international custodian bank), so all trades

are executed on an exchange. Both assumptions might have influenced our results. It is left to

future research to look at the welfare implications of horizontal and vertical integration in

securities trading and settlement from other angles.

Another critical aspect of our model is the assumption that, due to the preferences of investors, all

trades are cross-border trades, that is the buyer and the seller are located in different countries.

However, it is now easy to predict what kind of results one would get under alternative

assumptions. The opposite extreme would be to assume that all trades are domestic trades, that is

between investors located in the same country. Country A investors would have demand only for

A securities and country B investors only for B securities. In this case, the link would never be

used. Note that there would now be no competitive relationship between service providers in

different countries. Accordingly, horizontal integration of CSDs and complete separation would

lead to exactly the same results. However, vertical integration would lead to higher welfare than

separation and horizontal integration since all that matters is the fact that, in each country, the

exchange and the CSD offer complements. From these considerations, a simple lesson can be

learnt. If domestic investors have little interest in holding foreign securities, vertical integration of

domestic service providers may be desirable. If investors have instead strong preferences for

foreign securities, horizontal integration of CSDs may be the best from a welfare perspective.

Finally, it should be emphasised that we did not look at horizontal integration of exchanges. If we

had done that in our model, we would probably have come to the conclusion that exchanges

should not merge since they offer substitutes. However, there may still be good reasons why

exchanges should merge which are not taken into account in our model. For example, mergers of

exchanges lead to a concentration of liquidity. This liquidity concentration effect, however, does

not occur in our model.

30

Appendix A

In this appendix, the demand and supply functions of the investors for given transaction prices and

stock prices are determined. Each investor selects the alternative that gives him the highest benefit

according to the tables in Section 2. Note that in the benefits for the alternatives 2 to 5, the name i

of the respective investor does not occur, ie all investors located in the same country have the same

preferences over these four alternatives. Similarly, all investors located in the same country have

the same preferences over the alternatives 6 to 9. To begin with, we look at the country A investors

and have to consider several cases:

(1) u6i,A ≥ max{u7i,A, u8i,A, u9i,A}. First note that u6i,A ≥ max{u7i,A, u8i,A} ⇒ u6i,A ≥ u9i,A so that thelatter is redundant. Moreover, we easily find that u6i,A ≥ u7i,A ⇔ u4i,A > u5i,A and

u6i,A > u8i,A ⇔ u2i,A > u3i,A. Ie we have to distinguish two subcases:

(1a) u2i,A ≥ u4i,A. In this case, each investor selects either alternative 1 or alternative 2 or alternative6, ie we have to compare u1i,A, u2i,A and u6i,A:

u1i,A ≥ u2i,A ⇔ i ≥ rAA − pA ≡ α1u1i,A ≥ u6i,A ⇔ i ≥ rAA + 1− rBA

2− pA ≡ α2

u2i,A ≥ u6i,A ⇔ i ≥ 1− rBA − pA ≡ α3

Note that u2i,A ≥ u4i,A ⇔ rAA ≥ 1− rBA. It follows that α1 ≥ α2 ≥ α3. Ie all investors i ∈ [0, α3]choose alternative 6, all investors i ∈ [α3, α1] choose alternative 2 and all investors i ∈ [α1, 1]

31

choose alternative 1. From this, we get the supply and demand functions as follows:

SAA =

0, if α1 < 0

α1, if 0 ≤ α1 ≤ 11, if α1 > 1

DBA =

0, if α3 < 0

α3, if 0 ≤ α3 ≤ 11, if α3 > 1

SAB = 0, DBB = 0

Here, SAA is the supply of stock A in country A, SAB is the supply of stock A in country B and

DBA and DBB denote the respective demand for stock B.

(1b) u4i,A ≥ u2i,A. In this case, each investor selects either alternative 1 or alternative 4 or alternative6, ie we have to compare u1i,A, u4i,A and u6i,A:

u1i,A ≥ u4i,A ⇔ i ≥ 1− rBA − pA ≡ α3u1i,A ≥ u6i,A ⇔ i ≥ rAA + 1− rBA

2− pA ≡ α2

u4i,A ≥ u6i,A ⇔ i ≥ rAA − pA ≡ α1

Note that u4i,A ≥ u2i,A ⇔ 1− rBA ≥ rAA. It follows that α3 ≥ α2 ≥ α1. Ie all investors i ∈ [0, α1]choose alternative 6, all investors i ∈ [α1, α3] choose alternative 4 and all investors i ∈ [α3, 1]choose alternative 1. From that, we get exactly the same supply and demand functions as in case

(1a). Thus, we get

If u6i,A > max{u7i,A, u8i,A}, then

SAA =

0, if α1 < 0

α1, if 0 ≤ α1 ≤ 11, if α1 > 1

, DBA =

0, if α3 < 0

α3, if 0 ≤ α3 ≤ 11, if α3 > 1

SAB = 0, DBB = 0

32

As one would expect, SAA is increasing in rAA, DBA is decreasing in rBA and both functions are

decreasing in pA. Note that we do not necessarily have α1 = α3. It is immediate that α1 > α3 ⇔u2i,A ≥ u4i,A. Here, country A investors i ∈ [0, α3] choose alternative 6, ie sell stock A and buystock B in country A. A investors i ∈ [α3, α1] choose alternative 2, ie sell stock A in country Aand do not trade stock B. Finally, A investors i ∈ [α1, 1] choose alternative 1, ie do not trade atall. If instead α3 > α1, we find u4i,A ≥ u2i,A. In this case, all investors i ∈ [0, α1] choose alternative6, all investors i ∈ [α1, α3] choose alternative 4 and all investors i ∈ [α3, 1] choose alternative 1.

For the next two cases, we get the supply and demand in a very similar way:

(2) u7i,A > max{u6i,A, u8i,A, u9i,A}. We find that u7i,A > max{u6i,A, u9i,A} ⇒ u7i,A > u8i,A so that the latter

condition is redundant and get

If u7i,A > max{u6i,A, u9i,A} then

SAA =

0, if α1 < 0

α1, if 0 ≤ α1 ≤ 11, if α1 > 1

, DBB =

0, if α5 < 0

α5, if 0 ≤ α5 ≤ 11, if α5 > 1

DBA = 0, SAB = 0

with α5 ≡ 1− rBB − t − (pTB + pSB).

(3) u8i,A > max{u6i,A, u7i,A, u9i,A}. Here, we find u8i,A > max{u6i,A, u9i,A} ⇒ u8i,A > u7i,A and

If u8i,A > max{u6i,A, u9i,A} then

SAB =

0, if α6 < 0

α6, if 0 ≤ α6 ≤ 11, if α6 > 1

, DBA =

0, if α3 < 0

α3, if 0 ≤ α3 ≤ 11, if α3 > 1

SAA = 0, DBB = 0

with α6 ≡ rAB − t − (qA + qB)− (pTB + pSB).

33

For the last case, matters are somewhat more complicated:

(4) u9i,A > max{u6i,A, u7i,A, u8i,A}. This condition does not imply an order on the alternatives 2 to 5.We have to consider several subcases:

(4a) u2i,A > max{u3i,A, u4i,A, u5i,A}. In this case, each investor selects either alternative 1 oralternative 2 or alternative 9, ie we have to compare u1i,A, u2i,A and u9i,A:

u1i,A ≥ u2i,A ⇔ i ≥ rAA − pA ≡ α1u1i,A ≥ u9i,A ⇔ i ≥ rAB + 1− rBB − t − q − 2pB

2≡ α8

u2i,A ≥ u9i,A ⇔ i ≥ rAB + 1− rBB − rAA − t − q − 2pB + pA ≡ α9

Here, we have α1 + α9 = 2α8, ie α1 − α8 = α8 − α9. We have to consider two sub-subcases:

(4a1) α1 ≥ α8, ie α1 ≥ α8 ≥ α9. Here, all investors i ∈ [0, α9] choose alternative 9, all investorsi ∈ [α9, α1] choose alternative 2 and all investors i ∈ [α1, 1] choose alternative 1. From this, weget the supply function for stock A and demand functions for stock B as follows:

SAB =

0, if α9 < 0

α9, if 0 ≤ α9 ≤ 11, if α9 > 1

, SAA = α1 − α9, if α1 − α9 ≤ 1

1, if α1 − α9 > 1

DBB =

0, if α9 < 0

α9, if 0 ≤ α9 ≤ 11, if α9 > 1

, DBA = 0

(4a2) α8 ≥ α1, ie α9 ≥ α8 ≥ α1. Ie all investors i ∈ [0, α1] choose alternative 9, all investorsi ∈ [α1, α8] choose alternative 9; all investors i ∈ [α8, α9] choose alternative 1 and all investorsi ∈ [α9, 1] choose alternative 1. From this, we get the supply function for stock A and demand

34

functions for stock B as follows:

SAB =

0, if α8 < 0

α8, if 0 ≤ α8 ≤ 11, if α8 > 1

, DBB =

0, if α8 < 0

α8, if 0 ≤ α8 ≤ 11, if α8 > 1

SAA = 0, DBA = 0

The next three subcases are very similar:

(4b) u3i,A > max{u2i,A, u4i,A, u5i,A}. We consider the following two sub-subcases:

(4b1) α6 ≥ α10 ≡ 1− rBB − pB . We now get:

SAB =

0, if α6 < 0

α6, if 0 ≤ α6 ≤ 11, if α6 > 1

, DBB =

0, if α10 < 0

α10, if 0 ≤ α10 ≤ 11, if α10 > 1

SAA = 0 , DBA = 0

(4b2) α10 ≥ α6. We get the same result as in case (4a2).

(4c) u4i,A > max{u2i,A, u3i,A, u5i,A}.We consider the following two sub-subcases:

35

(4c1) α3 ≥ α11 ≡ rAB − rBB − t − q − 2pB + rBA + pA. We get

SAB =

0, if α11 < 0

α11, if 0 ≤ α11 ≤ 11, if α11 > 1

, SAA = 0

DBB =

0, if α11 < 0

α11, if 0 ≤ α11 ≤ 11, if α11 > 1

, DBA = α3 − α11, if α3 − α11 ≤ 1

1, if α3 − α11 > 1

(4c2) α11 ≥ α3. We get the same result as in the cases (4a2) and (4b2).

(4d) u5i,A > max{u2i,A, u3i,A, u4i,A}. We consider the following two sub-subcases:

(4d1) α5 ≥ α12 ≡ rAB − q − pB . We get

SAB =

0, if α12 < 0

α12, if 0 ≤ α12 ≤ 11, if α12 > 1

, DBB =

0, if α5 < 0

α5, if 0 ≤ α5 ≤ 11, if α5 > 1

SAA = 0, DBA = 0

(4d2) α12 ≥ α5. We get the same result as in the cases (4a2), (4b2) and (4c2).

Finally, there are several other cases we do not discuss in details but which are important and have

to be kept in mind. These are cases where all investors are indifferent between at least two

alternatives from 6 to 9 (or from 2 to 5). If for example u6i,A = u7i,A > max{u8i,A, u9i,A}, we find thatthe supply and demand may be given by any linear combination of the supply and demand

functions of the cases (1) and (2). The findings for other cases are similar.

36

Appendix B

¥Proof of propositions 1 and 2:

Proving these three proposition is very tedious, though not difficult. We therefore do not discuss

the entire proof in details, but rather look at a few cases only. Other cases can be dealt with in a

similar way. To derive the equilibrium stock prices and trading turnovers on the four stock

markets, we have to consider again several cases. We first consider cases in which both stocks are

traded:

(A) Assume that the conditions for case (1) in Appendix A are given, ie u6i,A > max{u7i,A, u8i,A}.We find SAA = DBA ≥ 0 and SAB = 0, DBB = 0. An equilibrium would require that country Binvestors choose DAA = SBA ≥ 0 and DAB = 0, SBB = 0. Immediate candidates for this are thefollowing:

(A1) Analogous to case (4a2) above, we know: If u9i,B > max{u6i,B, u7i,B, u8i,B},u2i,B > max{u3i,B, u4i,B, u5i,B} and α8 ≥ α1, where α1 ≡ rBB − pB and α8 ≡ rBA+1−rAA−t−q−2pA

2 , then

SBA =

0, if α8 < 0

α8, if 0 ≤ α8 ≤ 11, if α8 > 1

, DAA =

0, if α8 < 0

α8, if 0 ≤ α8 ≤ 11, if α8 > 1

SBB = 0, DAB = 0

37

An equilibrium would require SAA = DAA and DBA = SBA. Note that

α1 = α8 and α3 = α8⇔

rAA = 12− 14(t + q), rBA = 12 +

14(t + q),

xAA = xBA = 12 −14(t + q)− (pTA + pSA)

This is an equilibrium if 12 − 14(t + q)− (pTA + pSA) ≥ 0 and for rAA = 1

2 − 14(t + q) and

rBA = 12 + 1

4(t + q) all conditions above are fulfilled. It is easy to show that this leads after a fewconsiderations to the following result:

If

2− 4(pTA + pSA) ≥ t + q ≥ 2q +max{0, 4(pTA + pSA)− 4(pTB + pSB)}then

xAA = xBA = 12 −14(t + q)− (pTA + pSA), xAB = xBB = 0

is an equilibrium

(A2) Analogous to case (4b2) above, we know: If u9i,B > max{u6i,B, u7i,B, u8i,B},u3i,A > max{u2i,A, u4i,A, u5i,A} and α10 ≥ α6, where α6 = rBA − t − (qA + qB)− (pTA + pSA) andα10 = 1− rAA − (pTA + pSA), then

SBA =

0, if α8 < 0

α8, if 0 ≤ α8 ≤ 11, if α8 > 1

, DAA =

0, if α8 < 0

α8, if 0 ≤ α8 ≤ 11, if α8 > 1

SBB = 0 , DAB = 0

38

An equilibrium would require SAA = DAA and DBA = SBA. Note that

α1 = α8 and α3 = α8⇔

rAA = 12− 14(t + q), rBA = 12 +

14(t + q),

xAA = xBA = 12 −14(t + q)− (pTA + pSA)

This is an equilibrium if 12 − 14(t + q)− (pTA + pSA) ≥ 0 and for rAA = 1

2 − 14(t + q) and

rBA = 12 + 1

4(t + q) all conditions above (including the two conditions for case (1)) are fulfilled.However, the condition rBA − (qA + qB) ≥ 1− rAA implies (qA + qB) ≤ 0 which is in general notpossible.

(A3) Analogous to case (4c2) above, we consider the case that u9i,B > max{u6i,B, u7i,B, u8i,B},u4i,A > max{u2i,A, u3i,A, u5i,A} and α11 ≥ α3 (α11 and α3 defined in a similar way as before) and find

If 2− 4(pTA + pSA) ≥ t + q ≥ max{2q, 2q + 4(pTA + pSA)− 4(pTB + pSB),43q − 2

3+ 43(pTB + pSB)−

43(pTA + pSA)}

then xAA = xBA = 12 −14(t + q)− (pTA + pSA), xAB = xBB = 0

is an equilibrium

(A4) Analogous to case (4d2) above, we consider the case that u9i,B > max{u6i,B, u7i,B, u8i,B},

39

u5i,A > max{u2i,A, u3i,A, u4i,A} and α12 ≥ α5 (α12 and α5 defined in a similar way as before) and find

If (pTB + pSB) ≥ (pTA + pSA)2q ≤ t + q ≤ min{2− 4(pTA + pSA),

43q + 2

3+ 43(pTB + pSB)−

43(pTA + pSA)}

then xAA = xBA = 12 −14(t + q)− (pTA + pSA), xAB = xBB = 0

is an equilibrium

Summary of the cases (A1) to (A4):

Comparing the results for the cases (A1), (A3) and (A4), we can easily see that the conditions

given in (A1) are broader than those in (A3) and in (A4). Ie we can summarise these results as

follows:

If 2− 4(pTA + pSA) ≥ t + q ≥ 2q +max{0, 4(pTA + pSA)− 4(pTB + pSB)}then xAA = xBA = 12 −

14(t + q)− (pTA + pSA), xAB = xBB = 0

is an equilibrium

(A5) Analogous to case (4b1) above, we consider u9i,B > max{u6i,B, u7i,B, u8i,B},u3i,A > max{u2i,A, u4i,A, u5i,A} and α6 ≥ α10 and find that SAA = DAA and DBA = SBA would implyα6 ≤ α10 which is in contradiction to the conditions for case (A5).

(A6) Analogous to case (4d1) above, we consider u9i,B > max{u6i,B, u7i,B, u8i,B},

40

u5i,A > max{u2i,A, u3i,A, u4i,A} and α5 ≥ α12 and find

If t ≤ q ≤ min{1− 2(pTA + pSA), t + 2(pTB + pSB)− 2(pTA + pSA)}then xAA = 1− t2 − (pTA + pSA), xBA =

1− q2

− (pTA + pSA), xAB = xBB = 0is an equilibrium

(B) Assume that the conditions for case (2) of Appendix A and the analogous conditions for

country B investors are given, ie u7i,A > max{u6i,A, u8i,A, u9i,A} and u7i,B > max{u6i,B, u8i,B, u9i,B}. Wefind in a way similar to those used above

If t ≤ min{1− 2(pTA + pSA), 1− 2(pTB + pSB),q + 2(pTA + pSA)− 2(pTB + pSB), q + 2(pTB + pSB)− 2(pTA + pSA)}then xAA = 1− t2 − (pTA + pSA), xBB =

1− t2

− (pTB + pSB), xAB = xBA = 0is an equilibrium

(C) Assume that the conditions for case (3) of Appendix A and the analogous conditions for

country B investors are given, ie u8i,A > max{u6i,A, u7i,A, u9i,A} and u8i,B > max{u6i,B, u7i,B, u9i,B}. Wefind that this gives no equilibrium.

(D) Finally assume that the conditions for case (4) of Appendix A are given, ie

u9i,A > max{u6i,A, u7i,A, u8i,A}. This case is symmetric to case (A) above and leads to the following:

If 2− 4(pTB + pSB) ≥ t + q ≥ 2q +max{0, 4(pTB + pSB)− 4(pTA + pSA)}then xBB = xAB = 12 −

14(t + q)− (pTB + pSB), xBA = xAA = 0

is an equilibrium

41

If t ≤ q ≤ min{1− 2(pTB + pSB), t + 2(pTA + pSA)− 2(pTB + pSB)}then xBB = 1− t2 − (pTB + pSB), xAB =

1− q2

− (pTB + pSB), xBA = xAA = 0is an equilibrium

Note that the conditions for the first subcase in case (D) and those for the cases (A1) to (A4)

overlap partially. The same holds for the conditions for the second subcase in case (D) and those

for case (A6). Ie we may have multiple equilibria. In these cases, we apply or equilibrium

refinement as described in Section 2. If pA < pB , we select the equilibrium described in (A1) to

(A4) or (A6). If pA > pB , we select the equilibrium described in (D). If pA = pB , we select theallocation described in proposition 1. It is easy to show that this is an equilibrium allocation

though we have not discussed this equilibrium here.

It is easy to see that the results produced so far give us a proof of the propositions 1 and 2.

¥Proof of proposition 3:

(1) Assume q ≤ t and pSA = pSB . Note that under this assumption, pTA = pTB > 12cT cannot be the

case in equilibrium, because each exchange could do better by slightly decreasing its price and on

that way attracting all the trade. Now assume q ≤ t and pTA = pTB . Note that pSA = pSB impliesπ AS = [1−

12(t + q)− 2pA][pSA −

12cS + 12qA −

12cL] ≡ π1

while pSA < pSB implies

π AS = [1−12(t + q)− 2pA][2pSA − cS +

12qA − 12cL] ≡ π2

ie pSA = pSB > 12cS cannot be the case in equilibrium, because each CSD could do better by

slightly decreasing its price and on that way attracting all the settlement of exchange trades. Thus,

the following holds: If there is a symmetric equilibrium in which the link is used (and

pTA = pTB ≥ 12cT , p

SA = pSB ≥ 1

2cS), then it is characterised by pTA = pTB = 1

2cT , pSA = pSB = 1

2cS.

(2) Assume pTA = pTB = 12cT , p

SA = pSB = 1

2cS as given. Maximizing π1 with respect to qA under

the restriction q ≤ t and then assuming qA = qB gives qA = qB = min{12 t ; 23β + cL}. Thus, thefollowing holds: If there is a symmetric equilibrium in which the link is used, then it is

characterised by qA = qB = min{12 t ; 23β + cL}. Note that 12 t > 23β + cL ⇔ 4α < t − 2cL .

42

(3) We finally have to check under which conditions pTA = pTB = 12cT , p

SA = pSB = 1

2cS and

qA = qB = min{12 t ; 23β + cL} is indeed an equilibrium. It is clear that pTA = 12cT is a best response

of exchange A on pTB = 12cT , p

SA = pSB and q ≤ t . Now we look at CSD A and assume

pTA = pTB = 12cT , p

SB = 1

2cS and qB = min{12 t ; 23β + cL}.

First assume 12 t >23β + cL , ie t − 2cL > 4α. This implies qB = 2

3β + cL . Choosing pSA = 12cS and

qA = qB would give π AS = 19β

2. Now we show that no other strategy would give CSD A a higher

profit.

(i) Alternatively, CSD A could choose a response characterised by pSA < pSB and q ≤ t . To find thebest of such responses, we maximise π AS under pSA < pSB and q ≤ t . This is the same asmaximising π2 with respect to qA + 4pSA under qA + 4pSA ≤ t − 2

3β − cL + 2cS. Maximising π2without a constraint leads to the maximiser qA + 4pSA = 2

3β + cL + 2cS. This gives a profit ofπ AS = 1

9β2. Since the constrained maximisation cannot lead to a higher profit, choosing a response

characterised by pSA < pSB and q ≤ t can never be better than choosing pSA = 12cS and qA = qB .

(ii) Instead, CSD A could choose a response characterised by pSA > pSB and q ≤ t . We maximiseπ AS under pSA > pSB and q ≤ t , which is the same as maximising

π AS = [1−12(t + q)− cS − cT ][12qA −

12cL] ≡ π3

with respect to qA under qA ≤ t − 23β − cL . Maximising π3 without a constraint leads to the

maximiser qA = 23β + cL . This gives a profit of π AS = 1

9β2. Since the constrained maximisation

cannot lead to a higher profit, choosing a response characterised by pSA > pSB and q ≤ t can neverbe better than choosing pSA = 1

2cS and qA = qB .

(iii) Furthermore, CSD A could choose a response characterised by |pSA − pSB| ≤ 12(q − t) and

q ≥ t . Maximising π AS under |pSA − pSB| ≤ 12(q − t) and q ≥ t , ie maximising

π AS = [1− t − cT − 2pSA][pSA −12cS] ≡ π4

with respect to pSA gives pSA = 14[1− t + cS − cT ] and π AS = 1

8α2. It is easy to see that 18α

2 < 19β

2

if 12 t >23β + cL .

(iv) Next, CSD A could choose a response characterised by pSA − pSB ≥ 12(q − t) and q ≥ t .

43

Maximising π AS under pSA − pSB ≥ 12(q − t) and q ≥ t , ie maximising

π AS = [1− q − cS − cT ][12qA − 12cL] ≡ π5

with respect to qA under qA ≥ t − 23β − cL gives

qA = 1

6β + 12cL + 1

4 t , if109 β ≥ t − 2cL

t − cL − 23β, if

109 β ≤ t − 2cL

and

π AS = 1

2[16β + 1

4(t − 2cL)]2, if 109 β ≥ t − 2cL[β − 1

2(t − 2cL)][12(t − 2cL)− 13β], if

109 β ≤ t − 2cL

It is easy to see that this is smaller than 19β2.

(v) Finally, CSD A could choose a response characterised by pSB − pSA ≥ 12(q − t) and q ≥ t . We

have to maximise π AS under pSB − pSA ≥ 12(q − t) and q ≥ t , where

π AS = [2− t − q − 4pA][pSA −12cS]+ [1− q − 2pA][12qA −

12cL] ≡ π6

Unconstrained maximisation leads to pSA = 14[1− t + cS − cT ] > pSB = 1

2cS (and qA = 12 t − 1

3β),

ie violates the constraints so that it is clear that at least on constraint must be binding. Maximising

under pSB − pSA = 12(q − t) again leads to pSA = 1

4[1− t + cS − cT ], ie violates the constraintq ≥ t . Maximising under q = t leads to pSA = 1

3 − 512 t + 1

6cS − 13cT + 1

6cL which is smaller than

pSB = 12cS because t − 2cL > 4α. Ie this is the maximiser of π AS under pSB − pSA ≥ 1

2(q − t) andq ≥ t , and the maximum is given by π AS = 1

9β2.

This concludes the proof of part (1) of the proposition.

Now assume 12 t <23β + cL , ie t − 2cL < 4α. This implies qB = 1

2 t . Choosing pSA = 1

2cS and

qA = qB would give π AS = 14α[t − 2cL]. Again, we show that no other strategy would give CSD A

a higher profit.

(i) CSD A could alternatively choose a response characterised by pSA < pSB and q ≤ t . To find thebest of such responses, we maximise π AS under pSA < pSB and q ≤ t , ie maximising π2 with respectto qA + 4pSA under qA + 4pSA ≤ 1

2 t + 2cS. Unconstrained maximisation leads toqA + 4pSA = 1− 3

4 t − cT + cS + 12cL which is greater than

12 t + 2cS, because t − 2cL < 4α. Ie the

maximiser is qA + 4pSA = 12 t + 2cS and the maximum is π AS = 1

4α[t − 2cL]. Thus, choosing aresponse characterised by pSA < pSB and q ≤ t can never be better than choosing pSA = 1

2cS and

qA = qB .

44

(ii) Next, CSD A could choose a response characterised by pSA > pSB and q ≤ t . We maximisingπ AS under pSA > pSB and q ≤ t , ie π3 with respect to qA under qA ≤ 1

2 t . Unconstrained

maximisation leads to qA = 1− 34 t − cT − cS + 1

2cL which is greater than12 t , because

t − 2cL < 4α. Ie the maximiser is qA = 12 t and the maximum is π

AS = 1

4α[t − 2cL]. Thus,choosing a response characterised by pSA > pSB and q ≤ t can never be better than choosingpSA = 1

2cS and qA = qB .

(iii) Furthermore, CSD A could choose a response characterised by |pSA − pSB| ≤ 12(q − t) and

q ≥ t . Maximising π AS under |pSA − pSB| ≤ 12(q − t) and q ≥ t , ie maximising π4 with respect to

pSA gives π AS = 18α

2. Since 18α2 ≤ 1

4α[t − 2cL]⇔ 12α ≤ t − 2cL , we know now that

pTA = pTB = 12cT , p

SA = pSB = 1

2cS, qA = qB = 12 t can be an equilibrium only if

12α ≤ t − 2cL.

(iv) Instead, it could choose a response characterised by pSA − pSB ≥ 12(q − t) and q ≥ t .

Maximising π AS under pSA − pSB ≥ 12(q − t) and q ≥ t , ie maximising π5 with respect to qA under

qA ≥ 12 t gives

qA = 1

2α + 14 t + 1

2cL , if 2α ≥ t − 2cL12 t , if 2α ≤ t − 2cL

and

π AS = 1

8[α + 12 t − cL]2, if 2α ≥ t − 2cL

14α[t − 2cL], if 2α ≤ t − 2cL

It is easy to see that 18[α + 12 t − cL]2 is greater than 14α[t − 2cL], ie we know now that

pTA = pTB = 12cT , p

SA = pSB = 1

2cS, qA = qB = 12 t can be an equilibrium only if 2α ≤ t − 2cL.

(v) Finally, CSD A could choose a response characterised by pSB − pSA ≥ 12(q − t) and q ≥ t . Now,

we have to maximise π AS under pSB − pSA ≥ 12(q − t) and q ≥ t , where π AS = π6. Unconstrained

maximisation leads to pSA = 14[1− t + cS − cT ] (and qA = 1

4 t + 12cL), ie to p

SB − pSA < 0 so that it

is clear that at least on constraint must be binding. Maximising under pSB − pSA = 12(q − t) leads to

pSA = 14[1− t + cS − cT ], ie to pSB − pSA < 0 so that pSB − pSA ≥ 1

2(q − t) cannot be the onlybinding constraint. Maximising under q = t leads to pSA = 1

4 − 516 t + 1

4cS − 14cT + 1

8cL , which is

greater than pSB = 12cS because t − 2cL < 4α. Ie the maximiser of π AS under pSB − pSA ≥ 1

2(q − t)and q ≥ t is q = t and pSA = 1

2cS. The maximum is given by πAS = 1

4α[t − 2cL].

This concludes the proof of part (2) of the proposition.

45

¥Proof of proposition 4:

(1) Maximising π TA under |pA − pB| ≤ 12(q − t) and assuming that q ≥ t gives

pTA = 14[1− t + cT − 2pSA]. Maximising π SA under |pA − pB| ≤ 1

2(q − t) and q ≥ t givespSA = 1

4[1− t + cS − 2pTA]. Solving pTA = 14[1− t + cT − 2pSA] and pSA = 1

4[1− t + cS − 2pTA]gives pTA = 1

6(1− t − cS + 2cT ), pSA = 16(1− t − cT + 2cS). Ie if there is a symmetric equilibrium

in which the link is not used, then it is characterised by pTA = pTB = 16(1− t − cS + 2cT ) and

pSA = pSB = 16(1− t − cT + 2cS). We now only need to check under which conditions this is

indeed an equilibrium.

(2) Assume q ≥ t , pTB = 16(1− t − cS + 2cT ) and pSA = pSB = 1

6(1− t − cT + 2cS). If exchange Achooses pTA = 1

6(1− t − cS + 2cT ), it achieves a profit of πTA = 118α

2. Alternatively, exchange A

could choose a response characterised by pB − pA ≥ 12(q − t). However, if q is sufficiently high,

this cannot lead to a profit higher than 118α

2.

(3) Assume pTA = pTB = 16(1− t − cS + 2cT ), pSB = 1

6(1− t − cT + 2cS) and some qB ≥ 12 t . If

CSD A chooses pSA = 16(1− t − cT + 2cS) and qA = qB , then it achieves a profit of π SA = 1

18α2.

Alternatively, it could choose other responses. However, which responses are possible depends on

qB . If for example qB > t , CSD A cannot choose a response that is characterised by q ≤ t .Furthermore, if qB is sufficiently large, it is clear that choosing a response characterised by

pB − pA ≥ 12(q − t) cannot lead to a higher profit than 1

18α2. Similarly, if qB is sufficiently large,

it is clear that choosing a response characterised by pA − pB ≥ 12(q − t) cannot lead to a higher

profit than 118α

2 because such a response would imply that CSD A does not settle exchange trades

but makes profit only from the link which is hardly used because q is high. Thus, it is clear that

there always exist a number f such that all constellations with pTA = pTB = 16(1− t − cS + 2cT ),

pSA = pSB = 16(1− t − cT + 2cS) and qA = qB ≥ f are equilibria in which the link is not used.

¥Proof of proposition 5:

The proof is very similar to that of proposition 3 and therefore omitted.

¥Proof of proposition 6:

46

The proof is very similar to that of proposition 4 and therefore omitted.

¥Proof of proposition 7:

(1) We first determine the best response of the operator of the CSDs on pTA = pTB (under theconstraint pSA = pSB):

We first maximise π S under the constraint q ≤ t . Here, we haveπ S = [12 −

14t − 1

4(q + 4pSA)− pTA][q + 4pSA − 2cS − 2cL]

Maximising with respect to q + 4pSA gives q + 4pSA = 1− 12 t + cS + cL − 2pTA and

π S = 14[1−12t − cS − cL − 2pTA]2 ≡ π1S

We now maximise π S under the constraint q ≥ t . We haveπ S = [1− t − 2pSA − 2pTA][2pSA − cS]

Maximising with respect to pSA gives pSA = 14[1− t + cS − 2pTA] and

π S = 14[1− t − cS − 2pTA]2 ≡ π2S

Now note that π1S > π2S ⇔ t > 2cL . With that we get the following result:

If t > 2cL , we get as best responses all strategies that are characterised by

q + 4pSA = 1− 12 t + cS + cL − 2pTA and q ≤ t .

If t < 2cL , we get as best responses all strategies that are characterised by

pSA = 14[1− t + cS − 2pTA] and q ≥ t .

(2) We now prove part (1) of the proposition:

If q ≤ t , the exchanges enter into perfect Bertrand competition that leads to pTA = pTB = 12cT . Ie if

there is a symmetric equilibrium with trade that is characterised by q ≤ t , then it is characterisedby pTA = pTB = 1

2cT . Assume t > 2cL . The best response of the operator of the CSDs on

pTA = pTB = 12cT is according to (1) given by q + 4pSA = 1− 1

2 t + cS + cL − cT and q ≤ t . Finally,

47

it is easy to check that pTA = 12cT is a best response of exchange A on

q + 4pSA = 1− 12 t + cS + cL − cT , q ≤ t and pTB = 1

2cT .

(3) We now prove part (2) of the proposition.

Assume q ≥ t and some strategies pTB and pSA = pSB . As long as |pTA − pTB | ≤ 12(q − t), we have

πTA = [1− t − 2pTA − 2pSA][pTA −12cT ] ≡ πT,1A

Ie maximising πTA and then setting pTA = pTB is the same as maximising πT,1A which gives

pTA = 14[1− t + cT − 2pSA]. Ie if there is a symmetric equilibrium that is characterised by q ≥ t ,

then it is characterised by pTA = pTB = 14[1− t + cT − 2pSA]. Now assume t < 2cL . We know from

(1) that if there exists a symmetric equilibrium, it is characterised by pSA = 14[1− t + cS − 2pTA]

and q ≥ t . Ie we know that if there is a symmetric equilibrium, it is characterised bypTA = pTB =

1− t + 2cT − cS6

, pSA = pSB = 1−t+2cS−cT6 , q ≥ t .

To conclude the proof, we only have to show that exchange A cannot do better by deviating from

pTA = 1−t+2cT−cS6 , if pTB = 1−t+2cT−cS

6 and pSA = pSB = 1−t+2cS−cT6 if q ≥ h for some h ≥ t .

However, this is immediate since the only potential better response would be to select a response

characterised by pTB − pTA ≥ 12(q − t). But if q is sufficiently high, this would require a very small

pTA which may even lead to a negative profit for exchange A.

¥Proof of proposition 9:

The proof is very simple and therefore omitted.

¥Proof of theorem 10:

The proof is straightforward and therefore omitted.

48

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